Question

How do you solve the equation $\left| 2x+6 \right|=12$ .

Answer 1

Hint: Now we know that the function of modulus is defined as $\left| x \right|=x,x>0$ and $\left| x \right|=-x,x<0$ hence we will use the definition of modulus to and write the given equation for 2x + 6 > 0 and 2x + 6 < 0. Now we will solve the equation in both the intervals and hence find the solution of the given equation.

Complete step-by-step answer:
Now first let us see the definition of modulus function. The modulus function is the function which gives the absolute value of a number or a variable. It can also be called the distance function. Modulus of a given number or function is always positive.
The modulus function |x| is defined as
$\left| x \right|=x,x>0$ and $\left| x \right|=-x,x<0$ .
Now consider the given equation $\left| 2x+6 \right|=12$ . Let us use this definition to expand the modulus function.
\[\Rightarrow \left| 2x+6 \right|=2x+6,2x+6>0\] and \[\left| 2x+6 \right|=-2x-6,2x+6<0\]
Now simplifying the expression by taking 6 on RHS we get,
$\Rightarrow \left| 2x+6 \right|=2x+6,2x>-6$ and \[\left| 2x+6 \right|=-2x-6,2x<6\]
$\Rightarrow \left| 2x+6 \right|=2x+6,x>-3$ and $\left| 2x+6 \right|=-2x-6,x<-3$
Now in the interval $\left( -\infty ,-3 \right)$ we have,
$\Rightarrow \left| 2x+6 \right|=12$
Now since $\left| 2x+6 \right|=-2x-6,x<-3$ we have,
\[\begin{align}
  & \Rightarrow -2x-6=12 \\
 & \Rightarrow -2x=18 \\
 & \Rightarrow x=-9 \\
\end{align}\]
And in the interval $\left( -3,\infty \right)$ we have,
$\Rightarrow \left| 2x+6 \right|=12$
Now since $\left| 2x+6 \right|=2x+6,x>-3$ we have
$\begin{align}
  & \Rightarrow 2x+6=12 \\
 & \Rightarrow 2x=6 \\
 & \Rightarrow x=3 \\
\end{align}$
Hence the solutions of the equation are x = 3 and x = - 9.

Note: Now we can also solve this equation in a simpler manner. Now we know that the equation given is $\left| 2x+6 \right|=12$ Hence we get that 2x + 6 must be 12 or – 12. Hence we solve 2x + 6 = 12 and 2x + 6 = - 12 and find the value of x in both cases.